Exercise · ZW-POLYNOMIAL-RING-009 · ★★★ Advanced
Which of these polynomials is a unit of \(\mathbb{Z}_{4}[x]\)?
Choose one option.
Show answer
- B. \(1+2x\)
Explanation
We have \((1+2x)^2=1+4x+4x^2=1\) in \(\mathbb{Z}_{4}\), so \(1+2x\) is its own inverse. The other three have no inverse: each has a non-nilpotent coefficient or a non-invertible constant term.
Review the concept: Polynomial ring
The polynomial ring \(R[x]\) over a commutative ring \(R\) with identity consists of the finite sums of terms \(a_ix^i\), with coefficients \(a_i\in R\) and an indeterminate \(x\).
\[\Big(\sum_i a_ix^i\Big)\Big(\sum_j b_jx^j\Big)=\sum_k\Big(\sum_{i+j=k}a_ib_j\Big)x^k\]
More practice problems: polynomial ring
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- ZW-POLYNOMIAL-RING-005 Which statement about ℤ[x] is true?
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